Properties of the recursive divisor function and the number of ordered factorizations

The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive integer into $j$ factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated di...

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.

The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their connection with the powers moments of $|\zeta(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the lecture given at the conference ERDOS100 in Budapest, July 1-5, 2013.

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic f...

One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we e...

Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{rho} m(n) < exp[(log n)^{1/rho}/(loglog n)^{1+eps}] holds for all integers n>n_0, while, for a constant c>0, n^{rho} m(n) > exp[c(log n)^{1/\rho}/(loglog n)^{1/rho}] holds for infinitely many positive integers n, where rho=1.72864... is the real solution to zeta(rho)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^{\beta}$ and $G(n)^{\beta}$ up to $x$ for all real $\beta$ and the asymptotic bounds for $f(n)^{\beta}$ and $g(n)^{\beta}$ for all negative $\beta$.

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right). $$